# An Analog of the Salem-Zygmund Theorem for Orthogonal Polynomials on the Unit Circle

### Abstract

In this note, I study the result and proof of the classical Salem-Zygmund Theorem. I apply the method to random orthogonal polynomials on the unit circle. The goal is to find the distribution of *M*, the sup norm of suitably defined random polynomial orthogonal on the unit circle. In my proof, I use Bernstein and Chebyshev inequalities to achieve this goal. I find that for fixed large κ, the probability of *M* > κ drops significantly as the degree *n* of the orthogonal polynomial grows.

**Minnesota Journal of Undergraduate Mathematics**, [S.l.], v. 4, n. 1, feb. 2019. ISSN 2378-5810. Available at: <https://mjum.math.umn.edu/index.php/mjum/article/view/58>. Date accessed: 23 aug. 2019.